A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

1, 1, 1, 7, 1, 5, 1, 25, 7, 9, 1, 31, 1, 13, 5, 363, 1, 55, 1, 55, 7, 21, 1, 101, 7, 25, 33, 79, 1, 41, 1, 1335, 11, 33, 5, 305, 1, 37, 13, 177, 1, 59, 1, 127, 47, 45, 1, 1371, 7, 175, 17, 151, 1, 309, 7, 253, 19, 57, 1, 187, 1, 61, 67, 9923, 9, 95, 1, 199, 23, 113, 1, 927, 1, 73, 87, 223, 5, 113, 1, 2379, 715, 81, 1, 265, 11

Offset

1,4

Comments

The first negative term is a(840) = -445.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078898(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Mathematica

lpf[n_] := If[n == 1, 1, FactorInteger[n][[1, 1]]];
b[_] = 1;
A078898[n_] := A078898[n] = If[n == 0, 0, With[{t = lpf[n]}, b[t]++]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078898[n] - Sum[If[1 < d < n, f[d]*f[n/d], 0], {d, Divisors[n]}])]
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

Prog

(PARI)
up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to, n, A020639(n)));
A078898(n) = v078898[n];
A317833aux(n) = if(1==n, n, (A078898(n)-sumdiv(n, d, if((d>1)&&(d<n), A317833aux(d)*A317833aux(n/d), 0)))/2);
A317833(n) = numerator(A317833aux(n));

Crossrefs

Cf. A046644, A078898.

Cf. also A305798, A305803, A305804, A317830, A317834.

Sequence in context: A225459 A317847 A181722 * A021587 A195496 A065479

Adjacent sequences: A317830 A317831 A317832 * A317834 A317835 A317836

Keyword

sign,frac

Author

Antti Karttunen, Aug 10 2018

Status

approved